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The Trapezium Rule Formula

Part of Area Under Curves — GCSE Mathematics

This deep dive covers The Trapezium Rule Formula within Area Under Curves for GCSE Mathematics. Revise Area Under Curves in Graphs for GCSE Mathematics with 9 exam-style questions and 10 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 4 of 11 in this topic. Use this deep dive to connect the idea to the wider topic before moving on to questions and flashcards.

Topic position

Section 4 of 11

Practice

9 questions

Recall

10 flashcards

The Trapezium Rule Formula

Area ≈ h/2 × (y₀ + 2y₁ + 2y₂ + ... + 2yₙ₋₁ + yₙ)

Or in words: Area ≈ (h/2) × (first y + last y + 2 × all middle y-values)

Where:

  • h = equal width of each strip
  • y₀, y₁, ..., yₙ = y-values at equally spaced x-values (strip boundaries)
  • The first and last y-values appear once; all intermediate y-values appear twice

Example: Estimate the area under a curve using 4 strips of equal width h = 2, with y-values: y₀ = 3, y₁ = 5, y₂ = 8, y₃ = 7, y₄ = 4.

Area ≈ (2/2) × (3 + 2(5) + 2(8) + 2(7) + 4)

Area ≈ 1 × (3 + 10 + 16 + 14 + 4)

Area ≈ 1 × 47 = 47 square units

Keep building this topic

Read this section alongside the surrounding pages in Area Under Curves. That gives you the full topic sequence instead of a single isolated revision point.

The Trapezium Rule — How It WorksStep-by-Step ProcessMeasuring What You Cannot Calculate DirectlyArea Under a Curve — Core Principles

Practice Questions for Area Under Curves

On a velocity-time graph, what does the area under the curve represent?

  • A. The acceleration
  • B. The average velocity
  • C. The distance (or displacement) travelled
  • D. The speed at a given moment
1 markfoundation

Explain whether the trapezium rule gives an overestimate or underestimate for the area under the curve y = x² between x = 0 and x = 3. Justify your answer.

2 markshigher

Quick Recall Flashcards

What does the area under a speed-time graph represent?
Distance travelled. Area under a speed-time graph = distance travelled This works for any speed-time or velocity-time graph. Even if the graph is curved, the area still represents distance. Units: if speed is m/s and time is s, then area has units of metres (m).
What is the trapezium rule used for?
Estimating the area under a curve by dividing it into trapezium-shaped strips. Each strip is a trapezium (not a rectangle or triangle). The more strips used, the more accurate the estimate. It is an approximation — the actual area is slightly different because the tops of the strips are straight lines, not curves.

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The Trapezium Rule — How It Works

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Step-by-Step Process